Star Coloring of Graphs with Girth at Least Five

Abstract

For two disjoint vertex subsets X, Y of a graph G, we denote $$X \leftarrow Y$$XźY if every vertex of Y has at most one non-neighbour in X. A k-clique star partition of a graph G is $$V(G)=Q_1\cup Q_2\cup \ldots \cup Q_k$$V(G)=Q1źQ2źźźQk such that (i) $$Q_{i}$$Qi is a clique in G for all $$1\le i \le k$$1≤i≤k and (ii) $$Q_i\leftarrow Q_j$$QiźQj for all $$1 \le i < j\le k$$1≤i